# Full text of "Treatise On Analysis Vol-Ii"

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```244       XIV    INTEGRATION IN LOCALLY COMPACT GROUPS

If ju is any measure on G, we denote by fl the image of fi under the map-
ping xt-+x~1ofG onto G, so that we have

(14.1.4)                                   </,£> = </,/<>

for all /e tfC(G). It follows immediately from the definitions that we have
(j(s)/y = 60)/, each side being the function x^/Os""1*"1); and therefore,
for any measure /z on G, we have (y(s)/z)" = 8(s)fi. Hence if /z is a left-
invariant measure, # is a right-invariant measure, and vice versa.

(14.1.5)   Let G be a locally compact group. Then there exists a nonzero left-
invariant positive measure n on G, and every other left-invariant measure on G is
of the form a/i, where aeC.

(1) Existence. Let <#** denote the set of functions g e Jf R(G) which
are ^0 and not the zero function. For each/e «#*R(G) and each #e jf* ,
there exist positive real numbers cl9..., cr and elements sly..., sr in G such
that

(14.15.1)                          /^j>iY(*)0

r

(i.e., such that/(*)^ ]T cig(s^ix) for all xeG). For there exists a non-

i = l

empty open subset U in G such that a = inf g(x) > 0; since Supp(/) is com-

jceU

pact, there exist a finite number of points st e G (1 ^ / ^ r) such that the
jjU cover Supp(/), and then (14.1.5.1) is satisfied by taking q = ||/|| fa for
all i. We shall denote by (/: g) the greatest lower bound of the numbers

r

X Cj, for all systems (q,..., cr, slt..., sr) satisfying (14.1.5.1). The symbol

i=l

(/: g) has the following properties:

(i)   (Y(*)/:<7) = (/:<?)               for all  /e ^fR(G),   getfl,   seG;

(ii)   (af:g) = a(f:g)                for all  /eJfR(G),   geJtT*+>   a^O;

for all  A,f2eJl

(iv)   (f:g)^supf(x)/SMpg(x)    for all  /e JfR(G),   getf*;

xeG         xeG

(v)   (f:h)^(f:g)(9'K)           for all  /e JfR(G)   and   0,hintf*+;

for all  /,/0,^   inJT*,uality). (Consider first the case where / is bounded and of compact
```